Computer methods in applied mechanics and engineering Comput. Methods Appl. Mech. ELSEYIER Optimization Engrg. 149 (1997) 133-152 of multilaminated structures using higher-order deformation models Crist&%o M. Mota Soares”‘“, Carlos A. Mota Soares”, Victor M. Franc0 Correiab “IDMEC-Instiruto de Engenharia hENIDH-Escola N&rim Mecrinica-lnsriruto lnfunir Superior D. Henriquu, Tkxico, Av. Rovisco Pais, Av. Eng. Bormrville France, 1096 Lishm 2780 Oeiras, Codex, Portu,qcd Portugcd Abstract A refined shear deformation theory assuming a non-linear variation for the displacement field is used to develop discrete models for the sensitivity analysis and optimization of thick and thin multilayered angle ply composite plate structures. The structural and sensitivity analysis formulation is developed for a family of C” Lagrangian elements, with eleven, nine and seven degrees of freedom per node using a single layer formulation. The design sensitivities of structural response for static, free vibrations and buckling situations for objective and/or constraint functions with respect to ply angles and ply thicknesses are developed. These different objectives and/or generalized displacements at specified nodes, Hoffman’s or the volume constraints can be stress failure criterion, elastic strain energy, natural frequencies of chosen vibration modes, buckling load parameter semi-analytically. The accuracy and relative performance of the proposed discrete models are compared and discussed among the developed of structural material. The design sensitivities are evaluated elements and with alternative models. A few illustrative test designs are discussed to show the applicability either analytically or of the proposed models. 1. Introduction Laminated composite materials are being widely used in many industries mainly because they allow design engineers to achieve very important weight reductions when compared to traditional materials and also because more complex shapes can be easily obtained. The mechanical behavior of a laminate is strongly dependent of the fiber directions and because of that the laminate should be designed to meet the specific requirements of each particular application in order to obtain the maximum advantages of such materials. Accurate and efficient structural analysis, design sensitivity analysis and optimization procedures are very important to accomplish this task. Structural optimization with behavioral constraints, such as stress failure criterion, maximum deflection, natural frequencies and buckling load can be very useful in significantly improving the performance of the structures by manipulating certain design variables. Design sensitivity analysis is important to accurately know the effects of design variables changes on the performance of structures by calculating the search directions to find an optimum design. To evaluate these sensitivities efficiently and accurately it is important to have appropriate techniques associated to good structural models. It is well known that the analysis of laminated composite structures by using the classical Kirchhoff assumptions can lead to substantial errors for moderately thick plates or shells. This is mainly due to neglecting the transverse shear deformation effects which become very important in composite materials with low ratios of transverse shear modulus to in-plane modulus. This can be attenuated by Mindlin’s first-order shear deformation theory [ 1,2], but this theory yields a constant shear strain variation through the thickness and therefore requires the use of shear correction factors [3] in order to approximate the quadratic distribution in the elasticity theory. * Corresponding author 0045.7825/97/$17.00 P/I 0 S0045-7825(97)00066-2 1997 Elsevier Science S.A. All rights reserved 134 C.M.M. Soar-es et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-152 More accurate numerical models such as three-dimensional finite elements models can be used with adequate refined meshes in order to contemplate acceptable aspect ratios, but these models are computationally expensive. A compromising less expensive situation can be achieved by using single layer models, based on higher-order displacement fields involving higher-order expansions of the displacement field in powers of the thickness coordinate. These models can accurately account for the effects of transverse shear deformation yielding quadratic variation of out-of-plane strains and therefore do not require the use of artificial shear correction factors and are suitable for the analysis of highly anisotropic plates ranging from high to low length-to-thickness ratios. Pioneering work on the structural analysis formulation based in higher-order displacement fields can be reviewed in [4,5] where a theory which accounts for the effects of transverse shear deformation, transverse strain and nonlinear distribution of the in-plane displacements with respect to thickness coordinate is developed. Third-order theories have been proposed by Reddy [6-81, Phan and Reddy [9], Librescu [lo], Schmidt [ 111, Murty [ 121, Lenvinson [ 131, Seide [ 141, Murthy [ 151, Bhimaraddi and Stevens [ 161, Mallikarjuna and Kant [ 171 and Kant and Pandia [ 181 among others. Related overviews, closed form solutions and discrete models based on higher-order displacements fields can be found in Reddy [6,8,19], Mallikarjuna and Kant [ 171, Noor and Burton [20], Bert [21], Kant and Kommineni [22], Reddy and Robbins Jr. [23] and Robbins Jr. and Reddy [24] among others. Phan and Reddy [9] and Reddy and Phan [25] used a higher-order shear deformation theory based in a displacement field which accounts for layerwise parabolic distribution of transverse shear stress that satisfies the stress-free boundary conditions at the top and bottom surfaces of laminated plates, applied to the calculation of deflections, stresses, buckling loads and natural frequencies. Closed form solutions for simply supported laminated plates and a finite element model where the transverse displacement uses C’ continuity Hermite cubic shape functions to assure the continuity of the deflection and its derivatives across interelement boundaries were also analyzed. Other studies using the same displacement field are due to Reddy and Khdeir [26] and Khdeir [27], where closed form solutions are obtained and compared with classical plate theory (CPT) and higher-order shear deformation theories (HSDT) for unsymmetric cross ply rectangular composite laminates for buckling and free vibration under several boundary conditions. Finite element higher-order discrete models have been developed and discussed for vibration and/or buckling by Senthilnathan et al. [28], Kant et al. 1291, Putcha and Reddy [30], Kozma and Ochoa [31], Mallikarjuna and Kant [17], Liu [32], and Mallikarjuna and Kant [33]. Recently a four node rectangular element for the buckling analysis of multilaminated plates has been developed by Ghosh and Dey [34,35]. This model assumes a parabolic distribution of the transverse shear stresses and the non-linearity of the in-plane displacements across the thickness. The geometric stiffness matrix is developed using the in-plane stresses. An analytical solution based on a local high order deformation theory used for the determination of the natural frequencies and buckling analysis of laminated plates has been proposed by Wu and Chen [36]. The displacement fields in this theory are assumed to be piece-wise continuous high order polinomial series, layer by layer or sublaminate by sublaminate in thickness direction, accounting for the effects of transverse shear and normal deformation. Moita et al. [37] presented an eight node isoparametric discrete model based on an third-order expansion in the coordinate for the in-plane displacements and a constant transverse displacement. The model is applied to study several cases of composite plate and shell structures taking into consideration different number of layers, lamination angles, length-to-thickness ratios as well as symmetric and nonsymmetric laminates. The influence that the higher-order terms incorporated in the geometric stiffness matrix have on the prediction of the buckling load is also discussed. Related buckling studies are due to Kam and Chang [38] who developed a finite element plate model, based on the first-order shear deformation theory (FSDT) in which shear correction factors are derived from the exact expressions for orthotropic materials. Comprehensive overviews of buckling and vibration of composite plate-shell structures are given by Noor and Peters [39], Leissa [40,41], Kapania and Raciti [42,43] and Palazotto and Dennis [44]. Buckling analysis of moderately thick laminated plate and shell structures has been recently reviewed by Simitses [45] reporting works based on higher-order shear deformation theory and/or first-order shear deformation theory with or without a shear correction factor. Literature surveys in the field of sensitivity analysis and structural optimization such as those of Adelman and Haftka [46], Haftka and Adelman [47], Zyczowski [48] and Grandhi [49] reveal some lack of studies carried out on composite structures. Recently, Abrate [50] gave a wide perspective of work carried out by different researchers in the field of the optimum design of composite laminated plates and shells subjected to constraints on strength, stiffness, buckling loads and fundamental natural frequencies. Of the 84 papers reviewed, most of C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 1.3.3-152 135 them are based on variational approximation methods and the use of higher-order models is not mentioned. The optimal design of laminated plates for maximum buckling using discrete ply angles and a multiobjective approach to determine the optimal stacking sequence has been recently presented by Adali et al. [51,52]. Higher-order models have been used on the identification of material properties of multilaminated specimens using sensitivity analysis, optimization techniques and experimental vibration data enabling the identification of six mechanical properties [53-561. Csonka et al. [57] developed a higher-order axisymmetric discrete model where the nonlinear order theory is developed for the meridional displacement component through the thickness of the axisymmetric shell. The radial coordinate of a nodal point, ply thickness and the ply angle orientation of the fibers were used as design variables. The complexity of the displacement and strain fields was overcome through the use of symbolic computation to obtain the corresponding explicit expressions. The sensitivities of shape design of the element stiffness matrix and load vector were also obtained analytically through symbolic computation. The model was applied to the structural optimization of multilaminated axisymmetric shells for static type situations. Franc0 et al. [58] compared the use of models based on higher-order displacement fields to first-order and Kirchhoff models on the sensitivity analysis and optimization of laminated plates. In the present work a family of C” 9-node Lagrangian higher-order discrete models applied to static, eigenfrequency and buckling design sensitivity analysis and optimal design of multilaminated composite plates is presented. The design variables considered are the ply orientation angles of the fibers and the ply thicknesses. The design objectives are the minimization of generalized displacement components, minimization of structure elastic strain energy, maximization of natural frequencies of specified vibration modes, maximization of buckling load and/or minimization of structural volume or weight subjected to behavioral constraints. The sensitivities of static, eigenfrequency and buckling response with respect to the design variables can be evaluated either analytically or by using the semi-analytical technique [47]. The accuracy and relative computer efficiency of the developed higher-order models with respect to first-order models and classical plate theory based models are compared and discussed. Several examples are presented to illustrate the relative performance of the proposed models. 2. Higher-order displacement In order to approximate displacement components fields the three-dimensional elasticity problem to a two-dimensional laminate problem, the U, u and w at any point in the laminate space (Fig. 1) in the X, 4’ and z directions, Fig. 1. Laminate geometry and coordinate axes 136 C.M.M. Soores et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-152 respectively, are expanded in a Taylor’s series powers of the thickness coordinate function of X, y, z and t, where t is the time. The following higher-order displacement z. Each component fields are considered is a [17] HSDT 11 u=u,+zq+z*u~+z~(P~ v = v, + ze,, + z2v; + z’p; (1) w = W” + zq_ + z’wg HSDT 9 11 = U” + zq + z2u; + z’$Lq u = U” + ze, + z*v; + z+ (2) w = w,, HSDT 7 u = U” + zq + Z3qq v = u,, + zq + z$: (3) w = WC) where uO, v,, wO are the displacements of a generic point on the reference surface, CC),, fY are the rotations of normal to the reference surface about the y and x axes, respectively, and (o,, u:, v;S, wg, (p:, (4’1 are the higher-order terms in the Taylor’s series expansions, defined at the reference surface. 3. Laminate constitutive The constitutive represented by equations relations for an arbitrary ply k, written in the laminate (x, y, z) coordinate system can be where qi are the normal and shear stresses in an arbitrary point of kth lamina and E,, and r, are the normal and shear strains, respectively, in a arbitrary point in the laminate, referred to the laminate (x, y, z) coordinate system. The terms of constitutive matrix Qk of ply k are referred to the laminate axes (x, y, z) and can be obtained from the Q, matrix referred to the fiber directions (l-3) with the transformation 3, = T’Q,T (5) where T is the transformation 4. Higher-order finite element matrix from the (1,2, 3) fiber axes to the (x, y, z) laminate axes [59]. formulation The finite element formulation for the displacement field HSDT 11 represented by Eq. ( 1) applied to 9-node Lagrangian quadrilateral elements, will be briefly described. The strain components are given by au & xx =- ax au Yx, = yielding & =YY au ay + ax aV dy aW 52 au Y’z=z+ay az aw au x7 = aw z + yjy (6) C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) + 133-152 z3 137 (7) (8) A vector @,,, including the bending and membrane terms of the strain components the transverse shear terms can be defined in such a way as and a vector ET containing (9) (10) The strains can be written as [E,_,> r,, z,s; 11 Yr7 = (11) where Z,, and Z, are matrices containing powers of z coordinate (z” with n = 0, . . 3) defined in accordance with displacement fields and strain relations. Using C” Lagrangian shape functions [60] the displacements and generalized displacements defined in the reference surface are obtained within each element, respectively, by where the Z, matrix for the displacement field represented by Eq. (1) is given by -100zooz200230 z, = 0 -0 1 0 0 1 0 0 z 0 0 z 0 0 z2 0 0 z2 0 0 z3 01 N, are the Lagrange shape functions of node i and qr is the displacement displacement vector of the element, qr, by qe={-.q;.‘.}T, i=1,...,9 The strains in Eqs. (9) and (10) can be represented (14) vector of node i which is related to the (15) as (16) 138 C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-152 where B,, and B, are the strain-displacements matrices, respectively, for bending and membrane shear, relating the degrees-of-freedom of the element with the strain components. The Lagrangian functional for the eth element is and transverse (17) r,;}‘, ~1,= (u, u, w), VI: are the initial stress components, p is the material where E = {q,., <VY C? Y,~ r,, density, $ is the work done by the external forces, ti are the generalized velocities and V is the element volume. Applying Hamilton’s principle and adding the contributions of all finite elements in the domain one obtains (K + KJq + Mg = F (18) where K, M and K, are respectively the structure stiffness matrix, mass matrix and geometric stiffness matrix, q and 4 are the system displacement and acceleration vectors, respectively. This equation yields for static linear analysis, free harmonic vibrations and linear elastic buckling analysis, respectively Kq=F (19) Kq-w2Mq=0 (20) Kq + /U&q = 0 (21) where o represents the natural frequencies of the structure and A the buckling load parameter. K, M and K, matrices are obtained by assembling the corresponding element matrices K’, M’ and Kk of the finite elements. The stiffness and mass matrices of the element are evaluated, respectively, as (22) (23) where 3, is the constitutive matrix in the (x, y, z) laminate axes for kth ply, N is the matrix of the Lagrange shape functions and pk the material density of kth layer. NL is the number of layers of the laminate, h, is the vector distance from the middle surface of the laminate to the upper side of kth ply (Fig. 1). Finally, det J is the determinant of the Jacobian matrix of the transformation from (5,~) natural coordinates to element (I, y, z) coordinates. The element geometric stiffness matrix is given by (24) where B, is a strain-displacements matrix containing the shape functions and their derivatives matrix containing powers of the z coordinate, both defined in such a way that and Z, is a au au au -----au au au aw aw aw T --ax'ay'azvax'ay*az'ax? ay' az ax 1 ay 7 ax ’ ay The matrix o;, is the initial stress matrix for the kth ply containing all the six stress components obtained by solving using Eq. (19) and using constitutive relations Eq. (4) (25) previously C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 139 I.?.?-IS2 (26) 5. Finite element models The finite element model having the displacement field represented by Eq. (I), briefly described above, will be referred to as Q9-HSDT 11. The remaining elements whose displacement fields are given by Eqs. (2) and (3) are developed easily from this parent element by deleting the appropriate degrees of freedom leading respectively, to the finite element discrete models referred to as Q9- HSDT 9 and Q9-HSDT 7. The C” 9-node Lagrangian first-order discrete model, referred to as Q9- FSDT 5, can also be obtained by deleting all high order terms and introducing the shear correction factors on the transverse shear terms of constitutive matrix [59]. Two other finite element models are also available in the optimization package. One is the 3-node triangular shear flexible plate, based on Mindlin’s theory which was developed by Lakshminarayana et al. [61], usually referred to as TRIPLT, where the nodal degrees of freedom are the displacements and rotations ug, uO, M”~,,H,, 19,. and their first derivatives with respect to x and y. The other element is the 3-node triangular discrete Kirchhoff theory multilaminated plate-shell element [62], known in the literature as DKT, first developed for isotropic plates and shells by Batoz et al. [63] and Bathe et al. [64], respectively. The above described finite element models, corresponding degrees of freedom and number of nodes are shown in Table 1. 6. Sensitivity analysis 6.1. Statics For static type situations represented by the equilibrium equation (19) the sensitivities in the design variables, b, of a generic function sp,= p,(q, b) are evaluated by with respect to changes (27) where 4 is the vector of adjoint displacements obtained from aq (28) K4,=ag The sensitivities represented by Eq. (27) can be evaluated at the element level by (29) Table 1 Finite element models available for comparison purposes Finite element discrete model Number of nodes Q9-HSDT 11 Q9-HSDT 9 Q9-HSDT 7 Q9-FSDT 5 TRIPLT 9 9 9 9 3 DKT ‘I) The rotation U”’ VI,. w,,, e,, e,, el” 0: is used only for shell problems. 3 140 C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-152 where 4;’ is the adjoint displacements vector for element f, and E the set of elements for which the design variable bj is defined. The function pi can represent either the objective function of the optimization problem or a constraint equation. In the first case it can represent the structure elastic strain energy or a generalized displacement in a specified point. In the second case it can represent a maximum displacement constraint, a maximum stress constraint or the Hoffman’s failure criterion [59]. A constraint in a generalized displacement component can be represented by q = 6i/sman - 1 co (30) where C$ is the generalized displacement component and a,,,,, is the maximum constraint in a stress component can be represented by q = +&, allowed displacement. - 1 SO A (31) where 0; is the stress component and cr,,, is the maximum allowed value for that stress component. Hoffman’s first ply failure criterion, can be represented as a constraint equation by The (32) where u, , o;, Us are the normal stress components in the ( 1, 2,3) fiber axes, u,*, u223,u,, are the shear stress components, XT, YT, Z, are the lamina normal strengths in tension along the 1,2, 3 directions, respectively, R, S, T are the shear strengths in the 23, 13 and 12 planes, respectively, and Xc, Yc, Zc are the lamina normal strengths in compression along the 1,2, 3 directions, respectively. The stress components in the (1,2,3) fiber axes for the kth ply are obtained from f”, v2 g3 (T12 g23 531: = T -‘(a,( 2)) = (33) T-fek{:;}~e) The sensitivities of Hoffman’s failure criterion with respect to the design variables are obtained applying (29). The vector of adjoint forces 4;’ for an arbitrary ply of element 8 is obtained by differentiating Hoffman’s expression in order to q’ yielding Eq. the The derivatives au, /dq’ are evaluated analytically from the constitutive relations (Eq. (4)). The derivatives ae / abi are obtained in a similar way, by replacing the terms a?, / dq’ by ks / ab in Eq. (34). The derivatives au,/ab;are evaluated analytically from constitutive relations. If the function q represents the structure elastic strain energy s=lJ=;qTKq the sensitivities expression of the elastic strain energy with respect to design variables (35) are obtained at element level by the C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-152 141 dU -= db, (36) In the present work the design variables sensitivities dK’/ ab, are evaluated from are the ply angles and the ply thicknesses of the laminate and the (37) where the derivatives aD/db, are obtained either analytically or by using forward finite difference 1621. 6.2. Free vibrations The problem of free undamped the eigenvalue problem ([email protected])@,=0 vibrations of a finite element discretized linearly elastic structure is defined by n=l,...,N (38) where N represents the total number of degrees of freedom. The solution of this eigenvalue problem consists of N eigenvalues of and corresponding eigenvectors O,,, where w,, is the natural frequency of vibration mode n. For single eigenvalues, if we consider a vibration mode 0, corresponding to the natural frequency w,, normalized through the relation [email protected], = 1, the sensitivity of natural frequency with respect to changes in the design variable b, is obtained at the element level from 2 aW' --)o;’ p (39) ab, where 0;’ is the pth mode vector for element 8 and E is the set of elements defined. The sensitivities HI’ / db, are evaluated from for which the design variable b, is (40) where the derivatives d(_fi:_, ZT Z Wlab, are obtained either analytically or by using forward finite difference. In the case of multiple eigenvalues they are not, in general, differentiable with respect to design variables and Eq. (39) becomes inapplicable. This is because the eigenvectors corresponding to the repeated eigenvalues are not unique. In fact, any linear combination of the eigenvectors will satisfy the eigenvalue problem represented by Eq. (38). In this case the sensitivities can be obtained by calculating the directional derivatives [65-671 by considering a simultaneous change of all the design variables. 6.3. Buckling The buckling problem (K+h,K,)O,,==O of a finite element n= l,..., N discretized structure is defined by the eigenvalue problem (41) The solution of this eigenvalue problem consists of N eigenvalues A,, and corresponding eigenvectors O,,, where A,, is the buckling parameter corresponding to buckling mode n. For single eigenvalues, if we consider a buckling mode 0, corresponding to the buckling parameter AP, normalized through the relation O%K,Bp = 1, the sensitivity of the buckling parameter with respect to changes in the design variable b, is obtained at the element level from (42) 142 C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg 149 (1997) 133-152 The sensitivities dK’,/ ab, are evaluated by ~=ri~~~~~~,~(~~;~Z~[~ i ~-&&)]J&detJdidv (43) where the derivatives da;,/abi are evaluated by Eqs. (28)-(29). For multiple eigenvalues the sensitivities must be obtained by calculating 7. Optimal the directional derivatives ]65-671. design The structural optimization problem can be stated as min{ LI(b )} subject to: bi -C b, C by i = 1, . . , ndu $.(q,b)SO j= l,...) m (44) where L?(b) is the objective function, +,(q, b) are the m inequality constraint equations, bj and by are, respectively, the lower and upper limits of the design variables and ndv is the total number of design variables. The objective function can be the structural weight or volume, the generalized displacement in a given node, the elastic strain energy of the structure, the natural frequency of a given vibration mode or the buckling load parameter. The constrained optimization problems are solved by using the method of feasible directions [68] and the unconstrained problems are solved by using the BFGS (Broydon-Fletcher-Goldfarb-Shanno) method 1681. Ply angles and ply thicknesses can also be designed using a two level procedure [62,69,70] schematically min R subject to: i = I,...,ndv bi’ < bi I b; Design variables : Ply angles I min Q subject to: bf S bi I b/’ i = l,...,ndv j = l,...,m w~G-IM~ 0 Design variables : Ply thicknesses Fig. 2. Two-level optimization process. C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) II-IS2 143 described in Fig. 2. In this case the global objective is to minimize the weight/volume of the structure. At the first level of optimization, the weight /volume of the structure is kept constant and the optimal fiber directions are determined for minimum deflection, minimum elastic strain energy, maximum natural frequencies of given modes or maximum buckling load. At the second level the optimal ply thicknesses are found assuming the previously obtained ply angles. In this second level the weight/volume of the structure is minimized subject to structural behavior constraints as well as lower and upper bounds on design variables. 8. Numerical 8.1. Dejection applications and stresses in simply supported laminated plates The prediction of deflections and stresses of the present higher-order discrete models is compared with exact solutions obtained by Pagan0 and Hatfield [71] for a 3-Ply [0”/90”/0”] symmetric square (a X a) simply supported plate subjected to sinusoidal surface load q(x, y) given by q(x,y)=q,,sin($)sin(~), OSxCa, OGyCa, -h/2szch/2 The thickness of 1st and 3rd plies is h/4 each and the thickness of 2nd ply is h/2, where h is the total thickness of the laminate. The following material properties are used: E, = 25 X 10h psi, E, = E, = 10h psi, G,? = G, 3 = 0.5 X lo6 psi, G,, = 0.2 X 10h psi, v,~ = v,~ = ~23 = 0.25. The nondimensional maximum deflections of the 3-ply plate, for several length-to-thickness ratios (u/h), obtained with the present higher-order models are compared in Table 2 with the exact solutions of Pagan0 and Hatfield [71]. Alternative solutions obtained with discrete models based on first-order theory and discrete Kirchhoff theory (DKT) are also presented. A quarter plate with a 4x4 finite element mesh was used for all models. Percentage errors relative to Pagan0 and Hatfield results are shown, ranging from 1.3 to 4.4% in the case of higher-order models and achieving 7% and 10.6% in the case of first-order models for lower length-to-thickness ratios (a/h = 4). In the case of discrete Kirchhoff theory model it can be seen that the element is not applicable for length-to-thickness ratios lower than 50. The nondimensional stresses obtained with the present models are compared in Table 3 with the exact results from Pagan0 and Hatfield [71]. A good agreement was found for higher-order models even for very low a/h ratios where first-order models presented errors up to 50% for example in the prediction of normal stress v,,. This error decreases as the a/h ratio increases. The higher-order models have shown good results on shear transverse stresses for low a/h ratios, but for a/h ratios higher than 50 lower accuracy is achieved. Table 2 Maximum nondimensional deflections of 3.~1~ simply supported p = 4G,, + (E, + E2( 1 + 2u,,))/ square plates w.* = ~‘pl”l(l2(alh)“hy,,), (1 - VII1l?,) nlh 4 Pagano et al. [71] Q9-HSDT II Q9-HSDT 9 Q9-HSDT 7 Q9-FSDT 5 TRIPLT DKT Percentage IO 4.49 I 4.2894 (-4.4%) 1.709 I .6437 (-3.8%) 4.3545 (-3%) 4.3545 (-3%) I .6492 (3.5%) I .6492 (3.5%) 4.1739 (-7%) 4.0139 (- 10.6%) 0.9759 (-78%) I s975 (-6.5%) I s403 (-9.9%) 0.9759 (-43%) errors with respect to Pagan0 et al. [7l] solution. 20 SO 100 I. I 89 I .03I I.1615 (-2.3%) I.0157 (- 1.5%) 1.1626 (-2.2%) I.1626 (-2.2%) 1.1477 (-3.5%) I.1080 (-6.8%) 0.9759 (-18%) I.0159 (- 1.4%) 1.0159 (- I .4%) I.0134 (- 1.8%) 0.9789 (-5%) 0.9759 (-5.3%) I .OQ8 0.9944 (- 1.3%) 0.9945 (- 1.3%) 0.9945 (- 1.3%) 0.9939 (- 1.4%) 0.9596 (-4.8%) 0.9759 (-3.1%) 144 C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-1.52 Table 3 Nondimensional stresses of 3-ply simply supported ex alh 4 square plates (cr:otcrX) u: (a/2,a/2, +h/2) = 1/(4,(a/h)‘)(~~~“~~“), u:, (a/2,a/2, *h/4) * (~*;a:) = 14q,,Wh))(q,;~,J * (O,O, *h/2) ;0;;2,0,0) fd,o/2,0) 0.720, -0.684 0.725, -0.688 0.706, -0.706 0.706, -0.706 0.371/k0.371 0.352, -0.352 0.663, -0.666 0.616/-0.619 0.631/-0.631 0.6311-0.631 0.656, -0.656 0.625 / -0.625 -0.0467/0.0458 -0.0455/0.0446 -0.0461/0.0461 -0.0461/0.0461 -0.0333/0.0333 -0.0319/0.0319 0.292 0.249 0.249 0.249 0.194 0.161 0.219 0.212 0.214 0.214 0.155 0.135 Parano et Q9-HSDT Q9-HSDT Q9-HSDT Q9-FSDT TRIPLT al. [71] 11 9 7 5 10 0.559, 0.560, 0.562, 0.562, 0.488, 0.464, -0.559 -0.561 -0.562 -0.562 -0.488 -0.464 0.401, -0.403 0.389/-0.391 0.391/-0.391 0.3911-0.391 0.387, -0.387 0.367/-0.367 -0.0275lO.0276 -0.0272/0.0271 -0.027310.0273 -0.0273iO.0273 -0.0249/0.0249 -0.0237/0.0237 0.196 0.171 0.170 0.170 0.130 0.093 0.301 0.286 0.286 0.286 0.197 0.175 Pagan0 et Q9-HSDT Q9-HSDT Q9-HSDT Q9-FSDT TRIPLT al. [71] 11 9 7 5 20 0.543, 0.545, 0.545, 0.545, 0.525 / 0.497, -0.543 -0.543 -0.545 -0.545 -0.525 -0.497 0.308/-0.309 0.305/-0.305 0.305 / -0.305 0.305/-0.305 0.303/-0.303 0.288, -0.288 -0.0230/0.0230 -0.0229lO.0229 -0.0229/0.0229 -0.0229/0.0229 -0.0223/0.0223 -0.0212/0.0212 0.156 0.164 0.164 0.164 0.133 0.077 0.328 0.318 0.318 0.318 0.219 0.195 Pagan0 et al. [71] Q9-HSDT 11 Q9-HSDT 9 Q9-HSDT 7 Q9-FSDT 5 TRIPLT 50 0.539/-0.539 0.540, -0.540 0.540, -0.540 0.540, -0.540 0.536, -0.536 0.5061-0.506 0.276, 0.276, 0.276, 0.276, 0.275, 0.258, -0.0216/0.0216 -0.0215/0.0215 -0.0215/0.0215 -0.021510.0215 -0.0214/0.0214 -0.0202/0.0202 0.141 0.324 0.324 0.324 0.298 0.105 0.337 0.388 0.388 0.388 0.288 0.211 Pagan0 et al. [71] Q9-HSDT 11 Q9-HSDT 9 Q9-HSDT7 Q9-FSDTS TRIPLT -0.276 -0.276 -0.276 -0.276 -0.275 -0.258 8.2. Free vibration of simply supported square laminated plates In order to compare the accuracy on the prediction of natural frequencies of higher-order models (HSDT) and first-order models (FSDT) a few test cases are presented emphasizing the effect of the degree of orthotropy of individual layers (Young’s modules ratio E, /E,) and the effect of side-to-thickness ratios. The results obtained are compared with alternative solutions when available. To analyze the effect of the degree of orthotropy of individual layers we consider the free vibration problem of a simply supported laminated square plate having the following material properties: E, /E, = 3, 10, 20, 30 and 40; E, = E, = 10 GPa; G,,/E, = G,, /E, = 0.6; G,, /E, = 0.5; v,~ = v,~ = y3 = 0.25. The side-to-thickness ratio is a/h = 5. The following stacking sequences are considered: 4-ply [0”/90”/0”/90”], 6-ply [O”/9O”/O”/9O”/O”/9O”] and lo-ply [O”/9~/Oo/900/oO/900/O”/9~/Oo/90”]. A quarter plate 4 X 4 finite element mesh is used. Table 4 compares the results for the nondimensional fundamental natural frequency W obtained with HSDT models, FSDT models, discrete Kirchhoff model [62] (DKT) and a 3-D elasticity solution obtained by Noor [72]. It can be seen that the results obtained with DKT model are not acceptable with errors increasing for higher E, /E, ratios. The errors obtained with FSDT models increase for higher E, /E2 ratios achieving errors of about 5.8%. In Table 5 are presented the nondimensional natural frequencies corresponding to the symmetric modes higher than the fundamental one for the antisymmetric 4-ply [O”/90”/O”/90”] and also for a symmetric 5-ply [0”/90”/0” /90”/0’] lamination sequence assuming an E, /E2 ratio equal to 40 and a/h = 5. Discrepancies of FSDT models with respect to higher-order model Q9-HSDT 11 are presented. It can be observed that FSDT models accuracy on the prediction of natural frequencies decrease when higher vibration modes are considered achieving discrepancies of about 10% for the second vibration mode. C.M.M. Soar-es et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-1.52 Table 4 Nondimensional NL 4 fundamental Noor [72] Q9-HSDT I I Q9-HSDT 9 7 Q9-FSDT 5 TRIPLT DKT 6 Noor [72] Q9-HSDT I I QV-HSDT 9 Q9-HSDT 7 Q9-FSDT 5 TRIPLT DKT IO Noor [72] Q9-HSDT I I Q9-HSDT 9 Q9-HSDT 7 Q9-FSDT 5 TRIPLT DKT Percentage (w = w= Degree of orthotropy Model Q9-HSDT natural frequencies X IO) of symmetric of individual simply supported 145 square plates (a/h = 5) layers E, lE2 3 IO 2.6182 2.6059 (-0.47%) 2.5983 (-0.76%) 2.6004 (-0.68%) 2.6018 (-0.63%) 2.6047 (-0.52%) 3.0174 (15.2%) 3.2578 3.2594 (0.05%) 3.2513 (-0.20%) 3.2780 (0.62%) 3.2899 (0.99%) 3.2999 (1.29%) 3.7622 3.7871 (0.66%) 3.7793 (0.46%) 3.8502 (2.34%) 3.87.55 (3.01%) 3.8873 (3.33%) 5.2975 (40.8%) 4.0660 4.1094 ( I .07%) 4.1022 (0.89%) 4.2135 (3.63%) 4.2480 (4.48% j 4.2598 (4.77%) 4.2719 4.3289 ( I .33%) 4.3224 (1.18%) 4.4683 (4.60%) 4.5084 (5.54%) 4.5 I98 (5.80%) 7.0879 (65.9%) 2.6440 2.6287 (-0.58%) 2.621 I (-0.87%) 2.6224 (-0.82%) 2.6229 (-0.80%) 2.6240 (-0.76%) 3.0466 (152%) 3.3657 3.3546 (-0.33%) 3.3467 (-0.57%) 3.3622 (-0.11%) 3.3674 (0.05%) 3.3726 (0.21%) 3.9359 3.9342 (-0.04%) 3.9267 (-0.23%) 3.9673 (0.80%) 3.9772 ( I .05%) 3.9841 ( I .22%) 55364 (40.7%) 4.2783 4.2848 (0.15%) 4.2782 (-0.002%) 4.3420 ( I .49%x) 4.3532 (I .75%) 4.3607 ( I .93%) 4.509 I 4.5223 (0.29%) 45164 (0.16%) 4.6005 (2.03%) 4.6 I06 (2.25%) 4.6184 (2.42%) 7.46SS (65.6%) 2.6583 2.6409 (-0.66%) 2.6332 (-0.94%) 2.6338 (-0.92%) 2.6336 (-0.93%) 2.6337 (-0.92%) 3.0615 (15.2%) 3.4250 3.4066 (-0.54%) 3.3988 -0.76%) 3.405 I -0.58%) 3.4054 -0.57%) 3.4082 -0.49%) 4.0337 4.0 I76 (-0.40%) 4.0105 (0.57%) 4.0269 (-0.17%) 4.0256 (-0.20%) 4.0301 (-0.09%) 5.6556 (40.2%) 4.401 I 4.3880 (-0.30%) 4.3818 (-0.44%) 4.4075 (0.15%) 4.4024 (0.03%) 4.4078 (0.15%) 4.6498 4.6398 (-0.22%) 4.6344 (-0.33%) 4.6683 (0.40%) 4.6578 (0.17%) 4.6640 (0.30%) 7.6528 (64.6%) 20 30 40 errors with respect to Noor (721 solution. 8.3. Buckling of simply supported laminated plates The accuracy on the prediction of buckling loads of higher-order models (HSDT) and first-order models (FSDT) is analyzed and compared with alternative solutions for a simply supported square plate subjected to uniaxial membrane uniform compressive load lV1. The effect of the degree of orthotropy E, /E, is investigated. The following material properties are considered: E, /E, = 3, 10, 20, 30 and 40; E, = E, = 10 GPa; G,*/E2 = G,,fE, =0.6; G,,/E, =0.5; q2 = u,~ = vzj = 0.25. The side-to-thickness ratio is kept constant a/h = 10. A full plate 4 X 4 finite element mesh is used. The nondimensional buckling loads A for an anti-symmetric 4-ply [0”/90”/0”/90”] and for a symmetric S-ply [0”/90”/0”/90”/0”] are shown in Tables 6 and 7, respectively. The solutions obtained with HSDT and FSDT models are compared to a 3-D elasticity solution [36], a local higher-order deformation theory obtained by Wu and Chen [36] and a classical plate theory (CPT) [36]. The 146 C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-152 Table 5 Nondimensional E, IE, = 40 NL natural frequencies Model (w = wr ph /E2 X IO) of a simply Vibration Q9-HSDT I I Q9-HSDT 9 Q9-HSDT 7 Q9-FSDT 5 TRIPLT Q9-HSDT 1 I Q9-HSDT 9 Q9-HSDT 7 Q9-FSDT 5 5 TRIPLT Percentage Table 6 Nondimensional uniaxial Model buckling 2 12.0385 12.0559 12.8706 12.6612 (52%) 12.7586 (6.0%) 12.0385 12.0.559 12.8706 12.7115 (5.6%) 12.7.586 (6.0%) 4.5517 4.5s 19 4.55 19 4.5847 (0.6%) 45896 (0.7%) 1 I .5439 I I.5214 12.3709 12.6486 (9.6%) 12.7357 ( 10.3%) 13.4340 13.4316 13.Sll7 12.71 I5 (-54%) 12.8009 (-4.7%) 11 Q9-HSDT 9 Q9-HSDT 7 Q9-FSDT 5 3-D elasticity [36] Wu and Chen [36] CPT [36] Percentage Table 7 Nondimensional load A = N\u’l(f+‘) buckling 1I Q9-HSDT 9 Q9-HSDT 7 Q9-FSDT 5 3-D elasticity [36] Wu and Chen [36] CPT [36] Percentage for an anti-symmetric of individual 5 16.5854 16.5516 17.6423 17.3009 (4.3%) 17.5745 (6.0%) 20.7687 20.904 I 22.3941 2 I .3789 (2.9%) 2 I .6656 (4.3%) 20.7687 20.904 I 22.3941 21.3789 (2.9%) 21.6656 (4.3%) 17.1171 17.1091 17.7173 17.31 IO (1.1%) 17.5770 (2.7%) 19.9366 19.9179 21.4123 2 1.2922 (6.8%) 21.5693 (8.2%) 22.9358 22.9376 23.0846 2 I .4788 (-6.4%) 21.7609 (-5.1%) 6 4.~1~ [0”/90”/0”/90”] simply supported square plate layers, E, lEZ 10 20 30 40 5.1356 (-0.74%) 5.1308 (-0.83%) 5.1527 (-0.41%) 5.1575 (-0.32%) 5.1738 5. I739 5.5738 (7.7%) 9.0263 (0.1 I%) 9.0222 (0.06%) 9.1216 (1.2%) 9.1769 (1.8%) 9.0164 9.0176 10.2947 ( 14.2%) 13.7838 (0.29%) 13.7804 (0.27%) 14.0708 (2.4%) 14.2407 (3.6%) 13.7429 13.7461 16.9882 (23.6%) 17.8528 (0.39%) 17.8523 (0.39%) 18.3972 (3.5%) 18.7088 (5.2%) 17.7829 17.7886 23.6746 (33.1%) 21.3756 (0.45%) 2 I .3799 (0.47%) 22.22 14 (4.4%) 22.6889 (6.6%) 2 I .2796 2 I .2880 30.359 I (42.7%) solution. load A = N<a’l(Ezh’) for a symmetric Degree of orthotropy Q9-HSDT 4 3 errors with respect to 3-D elasticity Model ratio a/h = 5 and 11 model. Degree of orthotropy QB-HSDT 3 4.3289 4.3224 4.4683 4.5084 (4.1%) 45198 (4.4%) errors with respect to Q9-HSDT square plate with side-to-thickness mode I 4 supported 5-ply [0”/90”/0”/90”/0”] of individual stmply supported square plate layers, E, lE2 3 10 20 30 40 5.2816 (-0.82%) 5.2709 (-1.0%) 5.2807 9.9424 (-0.18%) 9.9230 (-0.37%) 9.9509 (-0.09%) 9.9417 (-0.18%) 9.9603 9.985 I 11.4918 (15.4%) 15.6716 (0.12%) 15.6418 (-0.07%) 15.7016 (0.31%) 15.6787 (0.12%) 15.6527 15.6934 19.7124 (25.9%) 20.5375 (0.35%) 20.5013 (0.17%) 20.5954 (0.63%) 20.5705 (0.51%) 20.4663 20.5 I76 27.9357 (36.5%) 24.7257 (0.54%) 24.6863 (0.38%) 24.8151 (0.9%) 24.8049 (0.86%) 24.5929 24.65 I7 36.1597 (47.0%) (-0.84%) 5.2801 (0.03%) 5.3255 5.3323 5.7538 (8.0%) errors with respect to 3-D elasticity solution C.M.M. Soares et al. I Compui. Methods Appl. Mech. Engrg. 149 (1997) 133-152 147 percentage errors with respect to 3-D elasticity solution are shown. The errors obtained with CPT solution increase for higher E, /E, ratios achieving 42.7%. The errors presented by FSDT model can be about 6% higher than the higher-order models Q9-HSDT 11 and Q9-HSDT 9 in the anti-symmetric case. In the symmetric case the results obtained with HSDT and FSDT models are both very good. 8.4. Optimal ply angles ,for maximum buckling load of rectangular plates A symmetric simply supported rectangular plate (a X 6) made of 4 plies of equal thickness with the lamination sequence [H/-0/-0/0] and having the following material properties: E, = 142.5 GPa, Ez = E, = = 0.25 and the total thickness h = 0.04 m is 9.79GPa, GIZ=G,,=4.72GPa, G,,= l.l92GPa, v,~=v,~=I/?~ now designed for maximum uniaxial buckling load N,. A linking relation between ply angles is imposed in order to have only one design variable. The angle 8 is measured counterclockwise from the x axis. Table 8 shows the optimal ply angle tl for maximum uniaxial buckling load, for several length-to-width ratios b/a and the corresponding value of the buckling load. A good agreement is found between all higher-order models. The t&t-order model shows some discrepancies lower than 2%. 8.5. Optimal design of square plate with central circular hole It is considered the optimal design of a simply supported laminated square plate with a central circular hole using a two level optimization process. The laminate is made up of 6 plies of equal thickness t, = 0.015 m and is subjected to an uniform pressure p, = 50000 N/m’. The side dimension of the plate is a = 2 m and the hole diameter is a/3. A full finite element model with 288 nodes and 64 elements is used (Fig. 3). At the first level of optimization the objective function is the minimization of the plate elastic strain energy and the design variables are the ply angles. The weight of the structure is kept constant. To accomplish this Table 8 Optimal ply angle B for maximum hlc1 I 1.2 I .3 I .‘l I .s I .6 1.7 I .8 2 Q9-HSDT uniaxial buckling load A’$of simply supported plates (u/h = 25) Q9-HSDT 7 Q9-HSDT 9 II rectangular Q9-FSDT 5 Ply angles N> (kN) PlY angles N! (kN) PlY angles N\ (kN) Ply angles NI (kN) 30.9” 45.5” 43.9” 42.3” 40.7” 39.4” 38.4” 38.2” 39.9” 9944.94 9745.04 9820.03 9863.92 9880.08 9872.3 I 9845.21 9806.64 9750.27 30.9” 45.5” 43.9” 42.2” 40.7” 39.4” 38.4” 38.2” 39.9” 9945.04 9735.06 98 IO.46 9854.65 9870.93 9863.17 9836.09 9797.64 9741.33 30.8” 45.1” 43.6 42.0” 40.5” 39.1” 38.1” 37.7” 39.3” 10027.9 I 9762.23 9841.18 9889.85 9910.61 9906.79 9882.49 9844.60 3 I .5” 45.5” 44.0 42.4” 40.9” 10194.58 9954.97 10030.57 10074.30 10089.24 10079.43 10050.04 10009.49 9949.84 9782.15 39.6 38.7” 38.5” 40.1” (b) Fig. 3. Square plate with central circular hole: (a) finite element mesh; (b) plate divided into 16 regions. C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-152 148 unconstrained optimization problem the plate is divided into 16 regions leading to 96 design variables (Fig. 3). The ply angles for all elements lying in one region are equal. In the initial design the ply angles are all set to 0“ (fibers aligned with x axis). At the second level of optimization the objective function is the minimization of the plate volume subject to Hoffman’s failure criterion with a stress safety factor of 2.5 and maximum deflection constraints with S,,, = 0.005 m. The design variables are the ply thicknesses. The thickness of each ply is assumed to be constant over the plate domain and the ply angles are kept constant at this optimization stage. The material properties are representative of those of a High-modulus Graphite/Epoxy with a fiber volume fraction of 0.6: E, = 220 GPa, E, = E, = 6.9 GPa, G,, = G,, = G,, = 4.8 GPa, v,~ = v,, = v,? = 0.25, p = 1640 Kg/m’. The strength properties are: X, = 760 MPa, YT= Z, = 28 MPa, Xc = 690 MPa, Yc = Z, = 170 MPa, R = S = T = 70 MPa. The upper and lower limits on the ply thicknesses are, respectively, 0.1 m and 0.001 mm. The optimal ply angles and ply thicknesses obtained with higher-order models and first-order model are presented in Table 9. Due to the symmetry of the optimal ply angles with respect to x and y axes, only the results for regions located on 1st quadrant are shown. A good agreement between all HSDT models is found and some discrepancies in the thickness distribution are obtained with the QPFSDT 5 model. Table 9 shows also the average CPU time ratios of HSDT models with respect to FSDT model. 8.6. Optimal design of cantilever panel The cantilever panel represented in Fig. 4 is designed for minimum volume subject to Hoffman’s stress failure criterion. The laminated panel is made up of 5 plies having the lamination sequence [ -20”25”/70”/25”/ -2O”]. The panel is subjected to an uniform pressure p; = 10 000 N/m*. The material properties are E, = 290 GPa, E,=E,=6.2 GPa, G,,=G,,=G23=4.8 GPa, u,,= y1 = v,, = 0.25, p = 1700 Kg/m’. The strength properties are: X, = 620 MPa, YT = Z, = 2 1 MPa, X, = 620 MPa, Yc = Z, = 170 MPa, R = S = T = 60 MPa. The objective is to minimize the volume of panel and find the optimal average thickness distribution subject to Hoffman’s first ply failure criterion. A strength safety factor of 2 is used. The panel is divided into 8 regions as shown in Fig. 4 and the thicknesses are constant inside each region. Thus, the optimization problem has 5 X 8 = 40 design variables. In the initial design all thicknesses in each ply are set to 12 mm. The upper and lower limits on the ply thicknesses are, respectively, 0.1 m and 0.001 mm. The optimal thickness distribution for regions 1, 2, 3 and 8 obtained with HSDT and FSDT models is presented in Table 10. It can be observed that a close agreement is obtained between all HSDT models but discrepancies of about 41% in the final volume are obtained with Q9-FSDT 5 model. In this example the lower accuracy in the stress prediction of FSDT model leads to an unsafe design. The average CPU time ratios with respect to FSDT model are also presented in Table 10. Table 9 Two-level optimization results for square plate with central circular hole Model Q9-HSDT Optimal ply angles for 1st quadrant [ -6.7”/ -0.9”], [-25.0”/-3.5”]. [-33.2”/ -4.7”]\ [-9.3”/-1.4”1\ [-6.7”/-0.9”]\ [-25.0”/-3.5”]\ [-33.2”/-4.7”]\ [-9.3”/-1.4”], [ -6.7”/ -0.9”]\ [ -6.6”/ -0.9”]\ [-25.0”/-3.5”]\ [-33.2”1-4.7”]\ [-9.3”/1.4”]> [-25.2”/-3.6”], [-33.7”/-4.8”]\ [-9.5”/-1.4”]\ Optimal thicknesses r,, t2,t,, I, (4 0.01435 0.00875 0.00967 0.01404 0.01512 0.00829 0.00829 0.01512 0.01512 0.00829 0.00829 0.01512 0.01193 0.01 153 0.01 153 0.01193 Initial volume (m’) Final volume (m’) 0.21906 0.17091 0.21906 0.17096 0.2 I906 0.17096 0.21906 0.17131 CPU time ratio 7.8 3.8 2.0 I II Q9-HSDT 9 Q9-HSDT 7 Q9-FSDT 5 149 C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) II-152 a Clamped 2 3 4 (a) Fig. Table panel: (a) finite element mesh and dimensions (m); (b) panel divided into 8 regions IO Average thickness Region distribution for minimum Ply I 8 Initial 4. Cantilever volume (m’) Final volume (m‘) CPU time ratio volume of a cantilever Average thickness Q9-HSDT II panel subiected distribution to Hoffman’s stress failure criterion (m) Q9-HSDT 9 Q9-HSDT 7 Q9-FSDT I 0.009766 0.009766 0.009766 0.008683 2 0.009754 0.009755 0.009754 0.00847 3 0.009756 0.009756 0.009754 0.008477 4 0.009758 0.009758 0.009754 0.008430 5 0.009769 0.009769 0.009766 0.008437 I 0.009466 0.009467 0.009466 0.007569 2 0.009455 0.009456 0.009454 0.007268 3 0.009456 0.009457 0.009454 0.007282 4 0.009457 0.009457 0.009454 0.007237 5 0.009468 0.009468 0.009466 0.007344 5 I I 0.009208 0.009208 0.009203 0.006 I4 I 2 0.009206 0.009206 0.009 0.006044 3 0.009201 0.009201 0.009199 0.006057 4 0.009195 0.009 0.009199 0.006044 5 0.009201 0.00920 0.009203 0.006133 I 0.00823 2 0.008256 3 0.0082 4 5 I I95 I I99 0.008230 0.008202 0.008255 0.0082 0.008217 0.00822 0.008181 0.008181 0.008225 0.001932 0.008173 0.008 0.008208 0.001949 0.360 0.360 0.360 0.360 0.270544 0.27053 0.270544 0.158477 5.6 3 I.8 I I8 172 I 0.00 1950 I9 I 0.001931 0.002278 150 C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-152 9. Conclusions A family of C” Lagrangian finite element models based on a refined shear deformation theory assuming a nonlinear variation for the displacement field has been developed. These models have been incorporated in an optimization package in order to obtain the structural sensitivities of response with respect to changes in design variables (ply angles and ply thicknesses) and to carry out the structural optimization of multilaminated plates, considering static, free vibration and buckling constraints and/or objective functions. The optimization process can be developed by using a two level scheme. Numerical illustrative applications have shown that higher-order models are able to accurately predict the behavior of highly anisotropic and/or low length-to-thickness ratio laminated plates with reasonable advantage over first-order models. Results presented in Tables 2 and 4 show that Kirchhoff models are unable to predict the behavior of highly anisotropic and/or low length-to-thickness ratio laminated plates. The quadrangular 9-node Lagrangian elements with higher-order displacement fields have shown good accuracy but the computational efficiency decreases when more complex displacement fields are used. First-order models have shown poor accuracy on stress prediction. Global constraints and/or objectives such as natural frequencies of specified vibration modes and buckling loads obtained with HSDT models are in better agreement with available 3-D elasticity solutions than the FSDT models (Q9-FSDT 5 and TRIPLT). Natural frequencies of free vibration modes higher than the fundamental one are better predicted by using the present refined HSDT models. The higher-order model with nine degrees-of-freedom per node Q9-HSDT 9 seems to represent a reasonable compromise between accuracy and computational efficiency. The analytical sensitivities in order to ply angles and ply thicknesses are easily and efficiently obtained for discrete models based on higher-order displacement fields. The use of FSDT models in optimization problems with first ply failure constraints can lead to unsafe designs. Higher-order theories are an improvement over the first-order theory in order to accurately predict the behavior of laminated plates with low length-to-thickness ratios and/or high degree of anisotropy. The use of higher-order discrete models in the optimal design of multilayered composite plates and sandwich plates is very promising as these models can be easily implemented in existing structural optimization packages. Acknowledgments Sponsorship from the following grants is gratefully acknowledged: Human Capital and Mobility Project ‘Diagnostic and Reliability of Composite Materials and Structures for Advanced Transportation Applications’ (Project CHRTX-CT93-0222) and Fundacao Luso-Americana para o Desenvolvimento (FLAD). References [I] R.D. Mindlin, Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. J. Appl. Mech., Trans. ASME 18 (1951) 31-38. [2] J.N. Reddy, A review of refined theories of laminated composite plates, Shock Vib. Dig. 22(7) (1990) 3-17. [3] J.M. Whitney, Shear correction factors for orthotropic laminates under static load, J. Appl. Mech. 40( 1) (1973) 302-304. [4] K.H. Lo, R.M. Christensen and E.M. Wu, A high-order theory of plate deformation. Part I: Homogeneous plates, J. Appl. Mech. (1977) 663-668. [5] K.H. Lo, R.M. Christensen and E.M. Wu, A high-order theory of plate deformation. Part 2: Laminated plates, J. Appl. Mech. ( 1977) 669-676. 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